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Show this is a projection on a vector space

  1. Feb 1, 2007 #1
    1. The problem statement, all variables and given/known data
    Let V=Mn(F) be the space of all nxn matrices over F; define TA=(1/2)(A+transpose(A)) for A in V.
    Verify that T is not only a linear operator on V, but is also a projection.


    2. Relevant equations
    A is a projection when A squared=A.


    3. The attempt at a solution
    I don't see how this works since clearly (1/2)(A+transpose(A)) squared does not equal (1/2)(A+transpose(A)) for all matrices.

    What am I doing wrong?
     
  2. jcsd
  3. Feb 1, 2007 #2

    AKG

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    A is a projection when A2 = A, not when (Ax)2 = Ax. So you don't need to look at whether

    [tex]\left [\frac{1}{2}(A + A^t)\right ]^2 = \frac{1}{2}(A + A^t)[/tex]

    You need to look at whether T2 = T, i.e. whether T(TA) = TA for all A, i.e. whether:

    [tex]\frac{1}{2}\left [\left (\frac{1}{2}(A + A^t)\right ) + \left (\frac{1}{2}(A + A^t)\right )^t\right ] = \frac{1}{2}(A + A^t)[/tex]

    Remember, you're used to writing A for your linear operators, and vectors in your vector space V are normally written as x or v or something. But now you have matrices AS THE VECTORS IN YOUR VECTOR SPACE, so you'll probably use A to stand for a vector, and now T is the operator.

    And you still need to check linearity.
     
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